arXiv subjects
Publications and source records attributed to Nicholas Kosar.
We consider the topology of real no $k$-equal spaces via the theory of cellular spanning trees. Our main theorem proves that the rank of the $(k-2)$-dimensional homology of the no $k$-equal subspace of $\mathbb{R}$ is equal to the number of facets in a $k$-dimensional spanning tree of the $k$-skeleton of the $n$-dimensional hypercube.
Recently, the homology and cohomology of non-k-overlapping discs, or, equivalently, no k-equal subspaces of Euclidean space, were calculated by Dobrinskaya and Turchin. We calculate the homology and cohomology of two classes of more general spaces: decreasing polychromatic configuration spaces and bicolored configuration spaces. Instead of all points behaving similarly, we allow for varying behavior between points.
Recently, Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant $c$ such that for any $k$ there is a family of graphs $G$ with $χ(G^k)$ unbounded and $χ_{\ell}(G^k)\geq c χ(G^k) \log χ(G^k)$. We also provide an upper bound, $χ_{\ell}(G^k)<χ(G^k)^3$ for $k>1$.